How do you find the exact functional value sin(405°+120°) using the cosine sum or difference identity?

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How do you find the exact functional value sin(405°+120°) using the cosine sum or difference identity?

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Answer 1 · 2015-08-21T03:06:33

Find: sin (405 + 120)

Ans: $\frac{\sqrt{6} - \sqrt{2}}{4}$ $(sqrt6 - sqrt2)/4$

Explanation:

sin (405 + 120) = sin (45 + 120) --> because 405 = 45 + 360 deg
Apply the trig identity: sin (a + b) = sin a.cos b + sin b.cos a
$sin 45 = \frac{\sqrt{2}}{2}$ $sin 45 = sqrt2/2$ ; and $cos 120 = - \frac{1}{2}$ $cos 120 = -1/2$
$sin 120 = \frac{\sqrt{3}}{2}$ $sin 120 = sqrt3/2$ ; and $cos 45 = \frac{\sqrt{2}}{2}$ $cos 45 = sqrt2/2$

$sin (45 + 120) = (\frac{\sqrt{2}}{2}) (- \frac{1}{2}) + (\frac{\sqrt{3}}{2}) (\frac{\sqrt{2}}{2}) =$ $sin (45 + 120) = (sqrt2/2)(-1/2) + (sqrt3/2)(sqrt2/2) =$
$= \frac{\sqrt{6} - \sqrt{2}}{4}$ $= (sqrt6 - sqrt2)/4$

Check by calculator: sin (45 + 120) = sin 165 = 0.26
$\frac{\sqrt{6} - \sqrt{2}}{4}$ $(sqrt6 - sqrt2)/4$ = 0.26 OK

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How do you find the exact functional value sin(405°+120°) using the cosine sum or difference identity?

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